A diesel engine in which an engine exhaust passage is branched to a pair of exhaust branch passages for purifying NO.sub.x, in which a switching valve is disposed at the branched portion of these exhaust branch passages to alternately guide the exhaust gas to one of the exhaust branch passages by a switching function of the switching valve, and in which a catalyst which can oxidize and absorb the NO.sub.x is disposed in each of the exhaust branch passages is well known (refer to Japanese Unexamined Patent Publication No. 62-106826). In this diesel engine, NO.sub.x in the exhaust gas introduced into one exhaust branch passage is oxidized and absorbed by the catalyst disposed in that exhaust branch passage. During this time, the inflow of the exhaust gas to the other exhaust branch passage is stopped and, at the same time, a gaseous reducing agent is fed into this exhaust branch passage. The NO.sub.x accumulated in the catalyst disposed in this exhaust branch passage is reduced by this reducing agent. Subsequently, after a short time, the introduction of the exhaust gas to the exhaust branch passage to which the exhaust gas had been introduced heretofore is stopped by the switching function of the switching valve, and the introduction of the exhaust gas to the exhaust branch passage to which the introduction of the exhaust gas had been stopped heretofore is started again.
However, when the introduction of the exhaust gas to a pair of exhaust branch passages is alternately stopped, the temperature of the catalyst in the exhaust branch passage on the side where the introduction of the exhaust gas was stopped is gradually lowered in the period where the introduction of the exhaust gas is stopped and is lowered to a considerably low temperature near the time when the introduction of the exhaust gas is started again. When the temperature of the catalyst becomes low in this way, there arises a problem in that the catalytic function of the catalyst is lowered, and therefore the oxidation and absorption function of NO.sub.x is not sufficiently carried out. In the period from when the introduction of the exhaust gas is started to when the catalyst temperature rises, the NO.sub.x is not absorbed by the catalyst and thus is discharged to the atmosphere.
Also, in this diesel engine, a pair of exhaust branch passages must be provided, and a switching valve becomes necessary. Therefore, the construction becomes complex. Further, the switching valve is always exposed to the high temperature exhaust gas, and therefore there arises a problem of durability of the switching valve. Also, from the viewpoint of the absorption of NO.sub.x, one catalyst is always idle, and therefore there is another problem such that the entire catalyst which is provided is not effectively utilized for the absorption of NO.sub.x.